Electric-Magnetic Duality Rotations in Non-Linear Electrodynamics

TL;DR

This paper explores a class of Lagrangians for Maxwell fields coupled to gravity that preserve electric-magnetic duality, linking them to solutions of the Hamilton-Jacobi equation in black hole backgrounds, with implications for string theory and higher-dimensional forms.

Contribution

It identifies a family of duality-invariant Lagrangians related to Hamilton-Jacobi solutions and examines their behavior in string theory contexts, including effects of axion and dilaton fields.

Findings

Born-Infeld Lagrangian is among the duality-invariant solutions.

Higher order electromagnetic terms break duality invariance in string theory.

Extension to 2k-forms in 4k dimensions discussed.

Abstract

We show that there is a function of one variable's worth of Lagrangians for a single Maxwell field coupled to gravity whose equations of motion admit electric-magnetic duality. Such Lagrangians are given by solutions of the Hamilton-Jacobi equation for timelike geodesics in Witten's two-dimensional black hole. Among them are the Born-Infeld Lagrangian which arises in open string theory. We investigate the effect of the axion and the dilaton in the open superstring case and we show that this theory loses its electric-magnetic duality invariance when one considers the higher order electromagnetic field terms. We discuss some implications for black holes in string theory and an extension to $2k$-forms in $4k$ spacetime dimensions.